Sodium and Potassium are not more or less Latin than Natrium and Kalium. (If anything, that K in Kalium is not particularly Latinate.) They are just alternate names proposed for the same element, just as Tungsten and Wolfram were.

Humphry Davy in Britain came up with Sodium and Potassium, while Ludwig Wilhelm Gilbert in Germany, a couple of years later, proposed Natronium (later Natrium) and Kalium.

None of the words are Latin in origin. In particular, Sodium comes from the Arabic suda “headache” (soda helps alleviate headaches), while Natr(on)ium derives from Natron, which is ultimately Ancient Egyptian. Potassium comes from English potash (ultimately Dutch), while Kalium comes from alkali < Arabic al-qalyah “plant ashes”.

The national breakdown of Sodium/Natrium and Potassium/Kalium divides Western and Central/Eastern Europe: countries culturally closer to Germany went with Gilbert’s German proposal (Dutch, Russian, Greek, and all countries in between), countries culturally closer to Britain used Davy’s original English proposal (French, Italian, Spanish).

EDIT: The question has been edited to ask about Silicium vs Silicon as well.

This time, Humphry Davy is behind the German name: he called it silicium, because he thought it was a metal. Nine years later, Thomas Thomson (chemist) called it silicon, because he thought it was a non-metal, like carbon and boron. It is in fact a Metalloid, which is in between.

This time, English is on its own: French, German, Italian, Spanish all call it silicium. Russian (and hence the Slavic languages) coined its own form kremnij from Ancient Greek krēmnos ‘cliff’. The other languages that use silicon are those under strong English influence: languages of the British Islands, and countries formerly under British or American colonialism: Philippines, Indonesia, Malaysia, Africa: